1. ## symmetry

Let G be the dihedral group of symmetries of the square (=D_4)

(a) what is the stabilizer of a vertex? an edge?
(b) G acts on the set of two elements consisting of the diagonal lines. What is the stabilizer of a diagonal?

2. Originally Posted by jin_nzzang
Let G be the dihedral group of symmetries of the square (=D_4)

(a) what is the stabilizer of a vertex? an edge?
Let 1, 2, 3, and 4 are vertices of the square (counterclockwise order) and e_1, e_2, e_3, and e_4 are edges between these vertices.
Let G_x be the stabilizer of x. Then,

G_1 = {e, (2,4) }, G_2 = {e, (1,3)}, ....
G_e1 = {e, (1,2)(3,4)}, G_e2 = {e, (14)(23)}, ....

(b) G acts on the set of two elements consisting of the diagonal lines. What is the stabilizer of a diagonal?
The stabilizer elements for the above action are identity, 180 degree rotation, diagonal flip, and 180 degree rotation after diagonal flip. What are these elements in D_4?